1. Introduction

Monolayer graphene (MG), constructed from a single layer of carbon atoms densely packed in a hexagonal lattice, was successfully produced by mechanical exfoliation [1, 2] and chemical vapor deposition [3–7]. This particular material constitutes an excellent system for studying two-dimensional (2D) physical properties, e.g., quantum Hall effects [8–11]. In the low-energy region |E^c,v| ≤ 1 eV, MG possesses isotropic linear bands crossing at the K (K′) point and is regarded as a 2D zero-gap semiconductor, where c (v) indicates the conduction (valence) bands [12]. The linear bands are symmetric about the Fermi level (E_F = 0) and become nonlinear and anisotropic at higher energy |E^c,v| > 1 [12]. Most importantly, the quasiparticles related to the linear bands can be described by the Dirac-like Hamiltonian [13] that is associated with relativistic particles and dominates the low-energy physical properties [11, 14, 15]. This special electronic structure has been verified by experimental measurements [2, 16].

MG has become a potential material candidate for nano-devices due to its exotic electronic properties. A good understanding of the behavior of MG under external fields [17–27] is useful for improving the characteristics of graphene-based nano-devices. In the presence of a uniform perpendicular magnetic field, the linear bands change into dispersionless Landau levels (LLs) which obey the specific relationship $E^{c,v}\propto \sqrt{n^{c,v}{B}_0}$, where n^c (n^v) is the quantum number of the conduction (valence) states and B₀ is the magnetic field strength. The related anomalous quantum Hall effects [11, 28, 29] and particular optical excitations [17] have been verified experimentally [10, 11]. The anisotropic behavior of dispersive quasi-Landau levels (QLLs) and the special selection rules presented in the related optical absorption spectra are shown for a modulated magnetic field. Furthermore, Haldane predicted the existence of quantum Hall effects in MG when it is subjected to the modulated magnetic field even without any net magnetic flux through the entire space [9]. For two cases of composite fields, a uniform magnetic field combined with a modulated magnetic field and a uniform magnetic field combined with a modulated electric potential, the LL properties are drastically changed by the modulated field [30–34]. The broken symmetry, displacement of the localization center, and alteration of the amplitude of the LL wave functions can, in some cases, be observed [35, 36].

MG is predicted to exhibit feature-rich optical absorption spectra. The spectral intensity is proportional to the frequency, but no prominent peak exists at low frequency [37]. However, a uniform perpendicular magnetic field can lead to a number of symmetric absorption peaks originating from LLs. Each peak obeys the specific selection rule Δn = |n^c − n^v| = 1, which has been confirmed by magneto-transmission measurements [16, 17, 38]. The selection rule is ascribed to the spatial symmetric configuration of the magneto-electronic wave function. Under a modulated magnetic field, the optical spectra exhibit many asymmetric principal peaks and subpeaks. The former satisfy a selection rule similar to those pertinent to LLs, while the latter do not [39, 40]. For a better understanding, it is necessary to investigate the primary features of optical excitations in composite magnetic fields with various field strength ratios between the uniform and the modulated fields.

The generalized tight-binding model with exact diagonalization method is developed to resolve the optical absorption spectra. By rearranging the tight-binding functions, the giant Hermitian matrix corresponding to the experimental fields can be transformed into a band-like matrix, so that the numerical computation time is greatly reduced [41–44]. Moreover, the π-electronic structure of MG can be solved in the wide energy range of 5 eV, a solution proven valid regardless of whether a magnetic, electric or composite field is applied. In this work, we focus on the optical absorption spectra of MG under a composite magnetic field. By means of controlling the ratio between a uniform magnetic field and a modulated one, the magneto-optical properties can be thoroughly explored and then presented in detail. For an increased modulated field, each symmetric peak, under a uniform magnetic field, splits into a pair of asymmetric peaks. Redshift is an obvious evolution exhibited by the threshold absorption frequency. Perceivably, each peak obeys the same selection rule as in the situation where only the uniform magnetic field exists until the modulated field is increased beyond a certain point. As a result, important differences of the absorption peaks are observed with regard to their structure, number, intensity, frequency and selection rule when the applied magnetic field is varied among the uniform, composite and purely modulated ones.

2. Band-like Hamiltonian matrix in external fields

The low-frequency optical properties of MG are determined by the π-electronic structure resulting from the 2p_z orbitals of the carbon atoms. The generalized tight-binding model with the exact diagonalization method is developed to characterize the electronic properties, and then the gradient approximation is applied to obtain the optical-absorption spectra. In the presence of magnetic fields, the vector potential induces Peierls phases [39, 41, 45, 46], which are then accumulated in the Bloch wave functions. Such a phase owns the extra period R or changes the unit cell. The enlarged rectangular unit cell, marked by the green rectangle in Fig. 1, is chosen as the primitive unit cell. In this work, the major discussions focus on R along the armchair direction. MG is subjected to three kinds of magnetic fields: a uniform perpendicular magnetic field, a periodically modulated magnetic field, and a composite magnetic field comprised of both the uniform and modulated parts. In the case of the uniform magnetic field B₀ = B₀ẑ, the period is given by $R=R_0={\varphi }_0/(3\sqrt{3}{b^{\prime }}^2{B}_0/2)$, where b′=1.42 Å is the C-C bond length, and ϕ₀ (= hc/e = 4.1356 × 10⁻¹⁵ [T/m²]) is the unit flux quantum. In the case of the modulated magnetic field, B_M = B_M sin(2πx/l_M) ẑ is exerted along the armchair direction, where B_M is the field strength and l_M is the period length with the modulation period R = R_M = l_M/3b′. In the case of the composite field, the period R = R_C is the least common multiple of R₀ and R_M. An enlarged rectangular unit cell induced by an external field encompasses 2R a atoms and 2R b atoms. The Hamiltonian matrix is a 4R × 4R Hermitian matrix spanned by 4R TB functions. Based on the arrangement of odd and even atoms in the primitive cell, the Bloch wave function |Ψ_k〉 can be expressed as:

 

Fig. 1 The primitive cell of a monolayer graphene in a uniform magnetic field and a spatially modulated magnetic field along the armchair direction.

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When electrons are excited from occupied valence to unoccupied conduction bands by an electro-magnetic field, only inter-π-band excitations exist in the zero temperature case. Based on Fermi’s golden rule, the optical absorption function results in the following form

3. Uniform magnetic field combined with a modulated magnetic field

Low-energy band structures of MG exhibit rich features in the presence of magnetic fields. The uniform perpendicular magnetic field causes the states to congregate and induces dispersion-less Landau levels (LLs), indicated by the red curves in Fig. 2 for B₀ = 4 T. The unoccupied LLs and occupied LLs are symmetric about the Fermi level (E_F = 0). Each LL is characterized by the quantum number n^c,v, which corresponds to the number of zeros in the eigenvectors of harmonic oscillators [42, 55]. Each LL is fourfold degenerate for each (k_x, k_y) state without considering the spin degeneracy. Its energy can be approximated by a simple square-root relationship $\left|E_n^{c,v}\right|\propto \sqrt{n^{c,v}{B}_0}$ [8, 56], which is valid only for $\left|E_n^{c,v}\right|\le 1\hspace{0.17em}\hspace{0.17em}\text{eV}$ [8].

 

Fig. 2 The low k_y-dependent energy bands for B_M ≤ B₀ case under (a) the uniform magnetic field B₀ = 4 T (red curves) and the composite field B₀ = 4 T in conjunction with R_M = 500 and B_M = 0.5 T (black curves), and (b) B₀ = 4 T in conjunction with R_M = 500 and B_M = 4 T. The triangular and circular symbols correspond to the band-edge states $k_{be}^{\alpha }$ and $k_{be}^{\beta }$, respectively.

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The main characteristics of the LLs are affected by the modulated magnetic field, as shown in Fig. 2(a) by the black curves for B_M = 0.5 T and R_M = 500. Only the n^c,v = 0 LL remains unchanged, i.e., it retains its fourfold degeneracy at E_F = 0. On the other hand, each dispersionless LL with n^c,v ⩾ 1 splits into two periodic oscillation subbands with double degeneracy. The subbands possess two kinds of band-edge states, $k_{be}^{\alpha }$ and $k_{be}^{\beta }$, which are associated with the minimum field strength B₀ − B_M and the maximum field strength B₀ + B_M, respectively. At a small modulation strength (B_M/B₀ ≤ 1/8), the subbands oscillate between two LLs at the field strengths B₀ ± B_M, as shown by the blue dashed lines in Fig. 2(a). The minimum and maximum of the oscillation subbands are proportional to $\sqrt{B_0-B_M}$ and $\sqrt{B_0+B_M}$, respectively. The surrounding electronic states at $k_{be}^{\beta }$ congregate more easily, which results in a smaller band curvature with a larger density of states (DOS). On the contrary, fewer states congregate at $k_{be}^{\alpha }$, a situation leading to the larger band curvature and a smaller DOS.

The band structure may be considerably modified by increasing the strength of the modulated magnetic field. As B_M increases to the magnitude of B₀, more complex energy spectra are introduced, as shown in Fig. 2(b) for R_M = 500 and B_M = 4 T. The oscillation subbands with n^c,v ⩾ 1 display wider oscillation amplitudes, stronger energy dispersions, and greater band curvatures. The strong oscillatory subbands with different quantum numbers overlap with one another, and the subband amplitudes are almost linearly magnified by B_M. The wave vectors associated with the band-edge states have no dependence on B_M. The greatest and smallest band curvatures occur at the local minimum $k_{be}^{\alpha }$ and the local maximum $k_{be}^{\beta }$, respectively. This implies that there are more $k_{be}^{\alpha }$ and fewer $k_{be}^{\beta }$ in the lower state energies. However, a simple relation between the subband amplitudes and $\sqrt{B_0\pm B_M}$ is absent.

An increase in the strength of the modulated field, as B_M ≫ B₀, can drastically change the band structure. Figure 3(a) illustrates the complex oscillatory subbands for R_M = 500 and B_M = 32 T. The oscillation amplitudes do not demonstrate a simple relationship with B_M. It should be noted that neither the minima of the conduction bands nor the maxima of the valence bands exceed E_F = 0. Thus, there is no overlap between the conduction and valence bands, regardless of the modulation strength. The wave vectors of the band-edge states in the case of a large modulation strength (B_M/B₀ ≥ 8) are different from those in the case of B_M ≤ B₀ (depicted in Fig. 2). Therefore, each type of band-edge state corresponds to two different energies. The band-edge states belonging to the α (β) type own the two energies $k_{be}^{\alpha -}$ and $k_{be}^{\alpha +}$ ( $k_{be}^{\beta -}$ and $k_{be}^{\beta +}$), as indicated by the orange and green triangles (blue and red circles) in the figure. Furthermore, the band structure in the composite fields displays band-edge state energies similar to those found under the influence of only a modulated magnetic field. In the presence of a pure modulated magnetic field, energy bands are partial flat bands at E_F = 0 and parabolic bands elsewhere, as shown in Fig. 3(b). Each parabolic subband has one specific band-edge state $k_{be}^{sp}$ at k_y = 2/3 and four extra band-edge states $k_{be}^{e+}$’s and $k_{be}^{e-}$’s at both sides of k_y = 2/3 (indicated by the open circle and triangles), respectively. For the former, the energy of $k_{be}^{sp}$ for the modulation strengths B₀ + B_M and B₀ − B_M is similar to those of $k_{be}^{\beta +}$ and $k_{be}^{\beta -}$, respectively. The latter $k_{be}^{e+}$ and $k_{be}^{e-}$ for the modulation strength B_M own the same energy as $k_{be}^{\alpha +}$ and $k_{be}^{\alpha -}$, respectively. The quantum number n^c,v corresponding to the partial flat bands at E_F = 0 is defined as zero, and n^c,v ≥ 1 are associated with the n-th (n = 1, 2, 3...) conduction and valence parabolic sub-bands. The definition of n^c,v is discussed in the following paragraph and graphically illustrated in Fig. 5.

 

Fig. 3 The low k_y-dependent energy bands for the B_M > B₀ case at (a) the composite field B₀ = 4 T in conjunction with R_M = 500 and B_M = 32 T, and (b) the pure modulated magnetic field R_M = 500, B_M =36, 32 and 28 T (red, black and blue curves, respectively). The triangular and circular symbols represent the same meanings as in Fig. 2.

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The above-mentioned features of band structures influenced by the modulated magnetic fields would be reflected in the density of states (DOS). In the presence of a uniform magnetic field, DOS exhibits many delta-function-like peaks resulting from LLs, as shown by the red curve for B₀ = 4 T in Fig. 4(a). Such peaks suggest that LLs possess 0D energy levels. With an applied modulated magnetic field, each peak, except for the n^c,v = 0 LL at E_F = 0, splits into a pair of square-root divergent peaks ω_α and ω_β, as shown by the black curve in Fig. 4(a) for R_M = 500 and B_M = 0.5 T. The square-root divergence implies that the 0D energy level becomes the 1D energy band. Each pair of peaks ω_α and ω_β corresponds to the $k_{be}^{\alpha }$ and $k_{be}^{\beta }$ states, respectively, and the peak frequencies are identical to those of LLs at B₀ + B_M and B₀ − B_M. However, the two peaks ω_α and ω_β possess different peak heights owing to the distinct band curvatures at the two kinds of band edge states. The frequency and intensity are strongly dependent on the modulation strength. The peak heights decline and the spacing between ω_α and ω_β rises for a large modulation strength B_M = 4 T, as shown in Fig. 4(b). This reflects the fact that the greater curvatures and wider amplitudes of the oscillation subbands are a result of the increasing B_M.

 

Fig. 4 The low-frequency density of states at (a) the uniform magnetic field B₀ = 4 T (red curves) and the composite field B₀ = 4 T in conjunction with R_M = 500 and B_M = 0.5 T (black curves), (b) B₀ = 4 T in conjunction with R_M = 500 and B_M = 4 T, and (c) B₀ = 4 T in conjunction with R_M = 500 and B_M = 32 T (blue curves), and the pure modulated magnetic field R_M = 500 and B_M =32 T (black curves).

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As the modulated field strength is further raised to B_M = 32 T (blue dashed curves in Fig. 4(c)), the DOS displays some features similar to those of the DOS in the modulated magnetic field where R_M = 500 and B_M = 32 T (black solid curves). The modulated magnetic field alone leads to a symmetric delta-function-like peak at ω = 0 (inset in Fig. 4(c)), as well as asymmetric square-root divergent peaks. The former comes from the partial flat bands at E_F = 0. The latter can be further divided into weak subpeaks ( ${\omega }_s^n$) and strong principal peaks ( ${\omega }_p^n$) that are, respectively, dominated by the extra band-edge states ( $k_{be}^{e+}$ and $k_{be}^{e-}$) and specific band-edge states ( $k_{be}^{sp}$). In the cases of a composite magnetic field, the frequencies and intensities of the subpeaks are almost the same as those in the modulated magnetic field alone. However, the principal peaks possess pair structures of ${\omega }_{P+}^n$ and ${\omega }_{P-}^n$, which respectively correspond to the two different band-edge states $k_{be}^{\beta +}$ and $k_{be}^{\beta -}$.

The LL wave functions exhibit a specific spatial symmetry. Associated with the odd and even atoms, the wave functions in Eq. 1 have only a phase difference of π, that is, $A_{\mathbf{o}}^{c,v}=-A_{\mathbf{e}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}=-B_{\mathbf{e}}^{c,v}$. Therefore, discussing only the amplitudes $A_{\mathbf{o}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}$ is sufficient for understanding the main characteristics of the LLs. The LL wave functions, which is the eigenvector of the simple harmonic oscillation, can be described by an n^c,v-th order Hermite polynomial multiplied by a Gaussian function, as shown in Fig. 5. These wave functions are distributed around the localization center, that is, at the 5/6 position of the enlarged unit cell. Similar localization centers corresponding to the other degenerate states occur at the 1/6, 2/6, and 4/6 positions. However, it is adequate to only consider any one center in evaluating the absorption spectra due to their identical optical responses. A simple relationship exists between the two sublattices of a- and b-atoms, i.e., $A_{\mathbf{o}}^{c,v}$ of n^c,v is linearly proportional to $B_{\mathbf{o}}^{c,v}$ of n^c,v + 1. Moreover, the conduction and valence wave functions are related to each other by $A_{\mathbf{o}}^c=A_{\mathbf{o}}^v$ and $B_{\mathbf{o}}^c=-B_{\mathbf{o}}^v$.

 

Fig. 5 The wave functions with n^c,v = 0 and n^c = 1 at $k_{be}^{\alpha }$ for (a)–(d) the uniform magnetic field B₀ = 4 T (red curves) and the composite field B₀ = 4 T together with R_M = 500 and B_M = 0.5 T (black curves), (e)–(h) B₀ = 4 T combined with R_M = 500 and B_M = 4 T, and (i)–(l) B₀ = 4 T combined with R_M = 500 and B_M = 32 T. The wave functions with n^c = 1 at k₁ (solid curves) and $k_{be}^{e\pm }$ (dashed curves) in the pure modulated field R_M = 500 and B_M = 32 T, are shown in (m)–(p).

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The modulated field has strong effects on the LL wave functions, but plays no role in the relationship between the valence and conduction states. The wave functions corresponding to $k_{be}^{\alpha }$, labeled in Fig. 3(a), are located at the minimum field strength B₀ − B_M. They exhibit slightly broadened and reduced amplitudes, as indicated by the black curves in Figs. 5(a)–5(d) for a small B_M = 0.5 T. However, the spatial symmetry of the wave functions remains unchanged. The simple relationship between $A_{\mathbf{o}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}$ of the wave functions is almost preserved. A stronger modulation strength results in greater spatial changes of the wave functions, as shown in Figs. 5(e)–(f) for B₀ = B_M = 4 T. An increased broadening and asymmetry of the spatial distributions at n^c,v = 0 are revealed. On the other hand, spatial distributions with n^c,v ⩾ 1 are only widened (i.e., n^c = 1 in Figs. 5(g) and 5(h)), whereas their spatial symmetry is retained.

With a large modulation strength B_M ≫ B₀, the wave function is characterized by a narrow distribution and an increased amplitude, as shown in Figs. 5(i)–5(l). For n^c,v = 0, the symmetry of the wave functions is almost restored. The amplitude $B_{\mathbf{o}}^{c,v}$, owing to the fact that it has the same number of zeros as $A_{\mathbf{o}}^{c,v}$, also contributes to the wave function. This means that carrier transfers occur between the a- and b-atoms. For n^c,v ≥ 1, the spatial symmetry is broken and the zero mode is changed. The wave functions at ${\text{k}}_{be}^{\alpha +}$ and ${\text{k}}_{be}^{\alpha -}$ are different combinations of two subenvelope functions, of which the location centers are at zero net field strength, as indicated by the green and orange curves, respectively. These features are also obtained in the pure modulated magnetic field case. The wave functions associated with k_y = k₁ for B_M = 32 T and R_M = 500 in Fig. 4(b) have two nearly overlapping subenvelope functions ϕ₁ and ϕ₀ with respect to 1 and 0 zero points, as shown in Figs. 5(m)–5(p) by the green and orange dashed curves. The quantum number n^c,v is defined by the larger number of zeros of the subenvelope functions. As the wave vector moves toward the extra band-edge states $k_{be}^{e+}$ and $k_{be}^{e-}$, the two subenvelope functions ϕ₁ and ϕ₀ shift to the center located at zero net field strength and overlap with each other. For example, the overlapping wave functions $A_{\mathbf{o}}^c$ correspond to $k_{be}^{e+}$ and $k_{be}^{e-}$ superpositioned by the different combinations of ϕ₀ + ϕ₁ and −ϕ₀ + ϕ₁, indicated by the green and red curves, respectively.

The low-frequency optical absorption spectrum of the LLs presents numerous features. Many delta-function-like peaks with an identical intensity exist, as shown in Fig. 6(a) for B₀ = 4 T by the red curves. A single peak ${\omega }_{LL}^{n{n}^{\prime }}$ comes from two equivalent transition channels: from the valence LL of n′^v (n^v) to the conduction LL of n^c (n′^c). The quantum numbers related to the transition between the LLs must obey the specific selection rule Δn = |n′^c(v) − n^v(c)| = 1. The main reason for this is that the velocity matrix M^cv has a non-zero value only when $A_{\mathbf{o}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}$ possess the same number of zeros and it is a dominant factor for the optical excitations of the prominent peaks. However, the modulated magnetic field modifies the number and intensities of the absorption peaks. The delta-function-like peak splits into two kinds of square-root-divergent peaks ${\omega }_{\alpha }^{n{n}^{\prime }}$and ${\omega }_{\beta }^{n{n}^{\prime }}$ with lower intensities, as shown by the black curves. The divergences of the former and the latter occur near the right- and left-hand sides of ${\omega }_{LL}^{n{n}^{\prime }}$. Each ${\omega }_{\alpha }^{n{n}^{\prime }}\hspace{0.17em}({\omega }_{\beta }^{n{n}^{\prime }})$ originates from the transitions of $k_{be}^{\alpha }(n^v)\to k_{be}^{\alpha }(n^c+1)$ and $k_{be}^{\alpha }(n^v+1)\to k_{be}^{\alpha }(n^c)$ [ $k_{be}^{\beta }(n^v)\to k_{be}^{\beta }(n^c+1)$ and $k_{be}^{\beta }(n^v+1)\to k_{be}^{\beta }(n^c)$], its absorption frequency is same as that generated from the LLs at B₀ − B_M = 3.5 T (B₀ + B_M = 4.5 T). This leads to a redshift for the threshold absorption frequency. The peak intensity of ${\omega }_{\alpha }^{n{n}^{\prime }}$ is lower than that of ${\omega }_{\beta }^{n{n}^{\prime }}$ since the DOS of $k_{be}^{\alpha }$ is weaker than that of $k_{be}^{\beta }$. These absorption peaks obey the selection rule Δn = 1, similarly to uniform magnetic field case. This is due to the fact that the simple relation between $A_{\mathbf{o}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}$ of the wave functions is almost preserved, even though the modulated field does have some effects on the wave functions.

 

Fig. 6 The low-frequency optical absorption spectra for the B_M ≤ B₀ case corresponding to (a) the uniform magnetic field B₀ = 4 T (red curve) and the composite field B₀ = 4 T together with R_M = 500 and B_M = 0.5 T (black curve) and (b) the composite field B₀ = 4 T combined with R_M = 500 and B_M = 4 T.

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As the modulated field strength B_M is increased to B₀ = 4 T, the frequency, intensity, peak number and selection rules in the optical absorption spectrum have evident variety, as shown in Fig. 6(b). The frequencies of ${\omega }_{\alpha }^{n{n}^{\prime }}$’s are largely reduced, but the opposite is true for ${\omega }_{\beta }^{n{n}^{\prime }}$’s. The fact that the α and β band-edge states, respectively, approach or depart from E_F = 0 is the main reason. This might lead to an abnormal relationship of absorption frequencies in that ${\omega }_{\alpha }^{n{n}^{\prime }}$’s with the large n^c,v’s occur at a lower frequency than ${\omega }_{\beta }^{n{n}^{\prime }}$ with the small n^c,v’s. For example, ${\omega }_{\alpha }^{12}$, ${\omega }_{\alpha }^{67}$ and ${\omega }_{\alpha }^{9\hspace{0.17em}10}$ are smaller than ${\omega }_{\beta }^{01}$, ${\omega }_{\beta }^{12}$ and ${\omega }_{\beta }^{23}$, respectively. The ratio between the intensities of ${\omega }_{\alpha }^{n{n}^{\prime }}$ and ${\omega }_{\beta }^{n{n}^{\prime }}$ is further diminished with increasing B_M. This can be clearly observed in transitions with small n^c,v’s, i.e., ${\omega }_{\beta }^{01}$ is ten times stronger than ${\omega }_{\alpha }^{01}$ in terms of the intensity. Moreover, more absorption peaks exist when B_M is increased. In addition to the peaks ${\omega }_{\alpha }^{n{n}^{\prime }}$ and ${\omega }_{\beta }^{n{n}^{\prime }}$ obeying the selection rule Δn = 1, two extra peaks with Δn = 2 and 3, ${\omega }_{\alpha }^{02}$ and ${\omega }_{\alpha }^{03}$, are generated. These two peaks reflect the fact that the wave functions of the LLs with n^c,v = 0 are drastically changed by the modulated magnetic field.

As the modulated field strength is further raised to B_M = 32 T (red dashed curves in Fig. 7), the absorption spectrum displays certain important features in the pure modulated magnetic field where R_M = 500 and B_M = 32 T (black solid curves in Fig. 7). When the absorption peaks are dominated by the modulated magnetic field, they can be divided into principal peaks ${\omega }_P^{nn+1}$ and subpeaks ${\omega }_s^{n{n}^{\prime }}$. These two kinds of peaks are not only seen in the modulated magnetic field case, but also appear in the composite magnetic field case as B_M ≫ B₀. The subpeaks ${\omega }_s^{n{n}^{\prime }}$ with the left-hand divergence arise from transitions between α type band-edge states (extra band-edge states) with the quantum number n and n′ in the B_M ≫ B₀ case (only B_M case). The association with the positions at zero net field strength are almost identical in both cases. However, we obtain two selection rules, Δn = 0 and Δn = 1, due to the complex overlapping behavior of two subenvelope functions in the wave function. On the other hand, the principal peaks ${\omega }_P^{nn+1}$ with right-hand divergence are much stronger than the subpeaks. They are attributed to the higher DOS consisting of the β type edge-states (specific band-edge states) with the quantum numbers n and n + 1 in the composite field B_M ≫ B₀ (pure modulated field). These absorption peaks obey the selection rule Δn = 1, similarly to the uniform magnetic field case. The principal peaks, however, possess a pair of peaks with ${\omega }_{P-}^{nn+1}$ and ${\omega }_{P+}^{nn+1}$, which respectively correspond to two different field strengths, |B₀ − B_M| = 28 T and |B₀ + B_M| = 36 T, and thus the difference between the two field strengths leads to distinct absorption frequencies. For B_M ≫ B₀, one can anticipate that the frequency discrepancy between the pair ${\omega }_{P-}^{nn+1}$ and ${\omega }_{P+}^{nn+1}$ becomes very small and eventually merges into a single peak, ${\omega }_P^{nn+1}$, i.e., the absorption spectrum is restored to the status of the pure modulated magnetic field case.

 

Fig. 7 The low-frequency optical absorption spectra for the B_M > B₀ case corresponding to the composite field B₀ = 4 T combined with R_M = 500 and B_M = 32 T (dashed blue curve) and the pure modulated magnetic field R_M = 500 and B_M = 32 T (black curve).

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The absorption frequency dependence on the modulated field strength is shown in Fig. 8 for R_M = 500 and B₀ = 4 T. In the range B_M ≤ B₀, the absorption peaks can be further divided into two categories based on the different field dependence. The frequencies of ${\omega }_{\alpha }^{n{n}^{\prime }}$ grow with increasing field strength, while the frequencies of ${\omega }_{\beta }^{n{n}^{\prime }}$ decline. Each of the absorption peaks ${\omega }_{\alpha }^{n{n}^{\prime }}$ and ${\omega }_{\beta }^{n{n}^{\prime }}$ has a linear relationship with B_M. This reflects the fact that the subband amplitudes are proportional to B_M within the range. It should be noted that the abnormal relationship of the absorption frequency, the intersection of ${\omega }_{\alpha }^{n{n}^{\prime }}$’s with large n^c,v’s and ${\omega }_{\beta }^{n{n}^{\prime }}$’s with small n^c,v’s, occurs at a sufficiently large modulated field. However, in the higher absorption frequency region or for the field range B_M > B₀, the linear relationship is broken since the subband amplitudes are no longer linearly magnified by B_M.

 

Fig. 8 The dependence of absorption frequencies ${\omega }_{\alpha }^{n{n}^{\prime }}$ and ${\omega }_{\beta }^{n{n}^{\prime }}$ with |Δn| = |n − n′| = 1 on the modulated strength B_M at B₀ = 4 T and R_M = 500.

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In addition to the modulated magnetic field, the modulated electric potential also induces rich features in the magneto-absorption spectra. In both cases, the four-fold degenerate LLs, except the n^c,v = 0 LL for the modulated magnetic field case, change into periodic oscillation subbands with double-degeneracy. The energy dispersions become stronger for an increased modulated field strength. With regard to the wave functions, the symmetry breaking of the LL spatial distributions is enhanced as the modulation potential or the quantum number grows. However, the asymmetry of the LL wave functions only at n^c,v = 0 is revealed when B_M is comparable to B₀. This implies that the optical absorption spectrum simultaneously exhibits the original peaks of Δn = 1 and the extra peaks of Δn ≠ 1. In the case of the modulated magnetic field, the extra selection rules, i.e., Δn = 2 and Δn = 3, come into existence only when B_M approximately equals B₀. On the other hand, extra selection rules induced by the electric potential would be generated when the modulation strength is increased. The extra peaks of Δn ≠ 1 are relatively easily observed at higher frequency.

4. Conclusion

The low-frequency optical absorption spectra in the presence of composite magnetic fields are studied by the generalized tight-binding model and gradient approximation. By means of controlling the ratio between B_M and B₀, systematic research on the magneto-optical spectra of MG can be thoroughly carried out. Under a uniform magnetic field, many delta-function-like peaks with uniform intensities exist in the optical spectrum. Each peak is generated from the LLs and satisfies the specific selection rule Δn = 1. At B_M ≪ B₀, each LL splits into two periodic oscillatory subbands with two kinds of band-edge states $k_{be}^{\alpha }$ and $k_{be}^{\beta }$, except for the LL with n^c,v = 0. Nevertheless, the spatial symmetry of the wave functions remains unchanged. The simple relationship between $A_{\mathbf{o}}^{c,v}$ and $B_{\mathbf{o}}^{c,v}$ of the wave functions is almost preserved. Optical spectra exhibit many pairs of square-root-divergent peaks ${\omega }_{\alpha }^{n{n}^{\prime }}$ and ${\omega }_{\beta }^{n{n}^{\prime }}$ that obey the selection rule Δn = 1. The two peak frequencies associated with $k_{be}^{\alpha }$ and $k_{be}^{\beta }$ are the same as those generated from the LLs at B₀ − B_M and B₀ + B_M, respectively. The implication is that a redshift occurs in the threshold absorption frequency. When B_M is increased to the magnitude of B₀, the oscillatory subbands display stronger energy dispersions and greater band curvatures. The strong oscillatory subbands with different quantum numbers overlap with one another. The wave functions exhibit broadened and reduced spatial distributions. In particular, the spatial symmetry of n^c,v = 0 is severely broken, which thus leads to the inclusion of extra peaks of Δn ≠ 1 in the absorption spectrum. Moreover, an abnormal relationship of absorption frequencies exists in the optical spectra, i.e., ${\omega }_{\alpha }^{n{n}^{\prime }}$’s with large n^c,v’s occur at frequencies lower than ${\omega }_{\beta }^{n{n}^{\prime }}$ with small n^c,v’s. As the modulated field strength is raised to B_M ≫ B₀, the optical spectra display certain features similar to those presented in the case of a pure modulated field B_M. Both cases contain the two selection rules Δn = 0 and Δn = 1, mainly owing to the complex overlapping behavior of two subenvelope functions in the wave function. However, in the case B_M ≫ B₀, the principal peaks consist of a pair of peaks that respectively correspond to the two different field strengths B_M ± B₀.

Very importantly, the generalized tight-binding model is developed to study monolayer graphene under various kinds of external fields. These fields can be uniform magnetic fields, modulated electric fields, modulated magnetic fields, and composite fields. The Hermitian Hamiltonian matrix used to determine the magneto-electronic properties becomes very large for the experimental field strengths. By means of rearranging the tight-binding functions, it is possible to transform this huge matrix into a band-like one to improve computational efficiency. The computation time can be further reduced by utilizing the characteristics of wave function distributions in the sublattices. In the generalized tight-binding model, the π-electronic structure of MG is exactly solved over the wide energy range ±5 eV, a solution that has proven to be valid even if a magnetic, electric or composite field is applied. Although the important interlayer interactions are not used in this case, the developed method is a generalized treatment more than just perturbations. This model is not only suitable in theoretical calculations of MG, but can also be extended to other stacked layer systems, i.e., AA-, AB-, ABC-stacked [53, 55, 57–59] and bulk systems [60]. The present study should prove very useful for comprehending other physical properties, such as Coulomb excitations [61–66] and transport properties [67–71].